There is a student in this class who has been in every room of at least one building on campus

Question

I am stuck on a question in Kenneth H. Rosen's Discrete Mathematics (7th edition):

There is a student in this class who has been in every room of at least one building on campus.

My solution is

$$\exists x \forall y \exists z B(x,y,z),$$

where $B(x,y,z)$ means "student $x$ is in room $y$ in building $z$", and the universe of discourse

• $x$: {all students in the class},

• $y$: {all rooms},

• $z$:{all buildings}.

Answer 1

Define:

$P(z,y)$ is ''z is in y"

$ Q(x,z)$ is "x has been in z"

$x$ ranges over students in this class

$y$ ranges over buildings.

$z$ ranges over rooms on campus

Statement:

There is a student in this class who has been in every room of at least one building on campus.

Using nested quantifiers, we can write it as:

$$ \exists x \ \exists y \ \forall z \ [ P(z, y) \rightarrow Q(x, z) ] $$